Question

A candle is 17 in. tall after burning for 3 hours. After 5 hours, it is 15 in. tall. Write a linear equation to model the relationship between height 'h' of the candle and time 't'. Predict how tall the candle will be after burning 8 hours.

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I'd like to attempt the last part on my own. I'm just having trouble writing a linear equation.
So far:

-I understand that it is burning at one inch an hour. For example, it was probably 16 inches at 4 hours. I gathered this from simple math (it has been two hours, and it has melted two inches.)
-I understand that there's a negative involved.

Answer 1

I don't have any ideas. I've had two but my gut tells me they're wrong. I'll provide if you want, though, but again, they're almost certainly incorrect.

Answer 2

@mathstudent55 (If you aren't busy, of course)

Answer 3

Oh, I was going to suggest using the slope formula and point-slope formula (using the hours and height as x and y) to find it.

Answer 4

Use the given info as two points, and write the equation of the line between them.

Answer 5

3 hours, 17 in.
5 hours, 15 in.

Use the points

(3, 17)
(5, 15)

Answer 6

Alright, so thatd be

\[m = \frac{ 17-15 }{ 3-5 }\]

Right?

Answer 7

Just making sure, I don't memorize formulas well.

Answer 8

Good.

Answer 9

Did you find the slope?

Answer 10

Which leaves me with a slope of

\[\frac{ 2 }{ -2 }\]

Answer 11

Correct, which reduces to?

Answer 12

-1, yes?

Answer 13

Correct.
Now you need to use the slope and one point to find the equation of the line.

Answer 14

\[y - 17 = -1(0 - 3)\]

Is this valid for finding the y-intercept for the equation??

Answer 15

I'd do this:

y = mx + b

Now plug in one of the known points into x and y, and plug in the slope into m.
Let's use point (3, 17)

17 = -1*3 + b

Now we can find the y-intercept, b.

Answer 16

Ohh.

That way I got -20.
The way I did it (using the slope-point formula) I got 20.
Judging by the graph (+ it'd make no sense in the context of a candle burning and all, esp. with the numbers given) I think it's the positive one.

Answer 17

I used the slope-intercept equation above.

You can also use the point-slope equation:

\(y - y_1 = m(x - x_1)\)

\(y - 17 = -1(x - 3) \)

You will see that they will give you the same equation.

Answer 18

b = 20

Answer 19

Leaving

y = -1x + 20

Answer 20

Correct

y = -x + 20

The want h for the height and t for time, so let's change to their variables:

h = -t + 20

Answer 21

As a quick check, let's make sure the given points work.

Check (3, 17)
h = -t + 20
17 = -3 + 20
17 = 17

Check (5, 15)
h = -t + 20
15 = -5 + 20
15 = 15

Yes, both points do work.

Answer 22

Now use the equation we got, and do the last part.
Here, you need time = 8, so let t = 8 in our equation
h = -t + 20

Answer 23

-8 + 20 = 12
Meaning after 8 hours of burning it's 12 inches tall.